On certain closed subgroups of .
On the surjectivity of Galois representations attached to elliptic curves over number fields
Bernoulli numbers, Hurwitz numbers, p-adic L-functions and Kummer's criterion.
Let K = Q(ζ) and let h be its class number. Kummer showed that p divides h if and only if p divides the numerator of some Bernoulli number. In this expository note we discuss the generalizations of this type of criterion to totally real fields and quadratic imaginary fields.
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